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Find the integer integers k for which there exists an integer x $\sqrt{39-6\sqrt{12}}+\sqrt{kx(kx+\sqrt{12}+3)}=2k$

So far I haven't advanced much. Removing brackets didn't do anything for me and was pretty bashy. I am actually lost on this problem, if only the three wasn't in the bracket....Any hint will be well appreciated

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  • $\begingroup$ Is there such a thing as a non-integer integer? What does the repetition mean? Can you find a minimum value of $k$? I would just plug in the smallest values of $k$ and see what happens. It might bring some enlightenment. $\endgroup$ Commented Jan 18, 2023 at 5:00

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Hint: Isolate $\sqrt{kx(kx+\sqrt{12}+3)}$ on one side of the equation, and square both sides. After a bit of simplification, you get an equation of the form $A(k,x) + B(k,x) \sqrt{3} = 0$, where $A(k,x)$ and $B(k,x)$ must be integers. This tells you that $A(k,x)$ and $B(k,x)$ must both be $0$.

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